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#

4. Summary and future work

We have reported a comparison of results obtained using our
relativistic partial wave and multipole calculation of the unpolarized
triply differential bremsstrahlung cross section with predictions of
the (Elwert-)Bethe-Heitler and Elwert-Haug approximations. Our
results have been summarized in Table 7.1. For the
high *Z* cases considered here and energies from 50-450 keV we find
that there is no significant region of systematic agreement with the
(Elwert-)Bethe-Heitler approximation for
.
At small
momentum transfers,
for forward emission and
with increasing momentum transfers for non-forward emission, the
Bethe-Heitler results tend to overestimate the exact partial wave
results. For larger momentum transfers they tend to underestimate the
exact partial wave results. We see differences ranging from factors
of two to factors of four, typically much larger than those seen in
the
results. The reduced differences in
are
attributed to cancellation of the differences in the small and large
momentum transfer regions when integrating over .
This
provides an example which confirms the expectation that observation of
provides a more stringent test of the underlying model.
In the case of
,
experimental uncertainty of less than
10-20% is required to differentiate results consistent with
Bethe-Heitler approximation, except in the tip region, (or 5-10% if
the Elwert factor is included) from those consistent with the exact
partial wave calculations. In the case of
measurements
can distinguish between the two theories even when experimental
uncertainty is as large as 30%. While use of the Elwert factor
typically improved agreement in
,
we find that in
it gives mixed results - causing larger differences at
small momentum transfers but improving the agreement somewhat at
larger momentum transfers.
In a similar comparison of the Elwert-Haug approximation with the
exact partial wave results for the unpolarized cross section for high
*Z* and energies from 50-450 keV, we find that at small momentum
transfers the Elwert-Haug results also tend to overestimate the exact
partial wave results. However, this overestimation is typically less
than 20% for the cases considered here; quite good agreement was
found in our calculations for *E*_{1}=300 keV with
keV at
low momentum transfers and small photon emission angles. We find that
for small emission and scattering angles the Elwert-Haug approximation
performs well at energies greater than 100 keV. For larger emission
angles or larger momentum transfers the Elwert-Haug results tend to
significantly underestimate the exact partial wave results,
approaching the Elwert-Bethe-Heitler results. We find that the
integrated cross section
(as reported by Tseng
[16] and Fink [18]) underestimates the
exact partial wave results at most energies, reflecting the
underestimate at large momentum transfers in
.

We now have available results, for coplanar and non-coplanar
geometries, over a broad range of atomic species and electron energies,
which we can compare with anticipated future experiments
[15]. Future theoretical efforts
could include examination of bremsstrahlung from electrons of lower or
higher energies than considered here. Lower energy cases would be of
particular interest in comparisons with predictions of so-called
atomic or polarizational bremsstrahlung [19]. At
higher energies one should see the nature of convergence toward high
energy limit forms. We should also note that limitations in our
current computer program have prevented us from considering cases
where the photon energy was low compared to the incident electron
energy (
*k*/*T*_{1} < 0.35). Extension of our code to include such cases
could be of interest for comparisons with future experiments.

** Next:** 5. Acknowledgments
** Up:** Comparison of relativistic partial
** Previous:** 3.2 Comparison with the
*Eoin Carney*

*1999-06-14*